Rigidity for critical metrics of the volume functional

Abstract

Geodesic balls in a simply connected space forms ${\mathbb{S}}^n$, ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$ are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary of the manifold is an Einstein hypersurface. In the same spirit we also extend a rigidity theorem due to Boucher et al. \cite{Bou} and Shen \cite{Shen} to $n$-dimensional static metrics with positive constant scalar curvature, which provides another proof of a partial answer to the Cosmic no-hair conjecture previously obtained by Chruściel \cite{Chrus}.

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References in corpus (1)

• Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

Cited by in corpus (1)

• Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary